General Continuum Mechanics

In the previous chapters, we discussed the concepts of strain and stress without focusing on any particular material. In this chapter, we examine a linear elastic materia1. The nonlinear elastic material (nonlinear elasticity) is discussed in Chap. 7. By elastic we mean that the material may be deformed linearly or nonlinearly but will return to its original configuration on release of the applied loads. On the contrary, plastic materials may not return to their original position on the release of applied loads. In another case, the material known as viscoelastic exhibits time-dependent properties. Plasticity and viscoelasticity with such inelastic behavior are treated in Chap. 7.

In this chapter we begin with constitutive equations of linear elastic solids, followed by Navier equations, energy principles, and the thermodynamics of solids (Love, 1934; Truesdell and Toupin, 1960; Little, 1973). Thermomechanically coupled equations of motion and heat conduction are shown to emerge as a result of the thermodynamic principles applied to elastic solids by use of the first and second laws of thermodynamics. Finally, mechanics of fiber composite materials is discussed as a part of linear elasticity.

Constitutive Equations for Linear Elastic Solids

Three-Dimensional Solids

In the theory of linear elasticity, we are concerned with an ideal material governed by Hooke's law. This law was proposed by Robert Hooke in 1678 in his essay “Ut tensio sic vis,” stating that “the power of a springy body is in the same proportion as the extension.”